"They want to make sure [their kids] are not exposed to things that might be hurtful," says a source close to the couple
“This is a natural evolution for Gwyneth and Chris as friends and as a couple,” a source close to the stars tells PEOPLE. “They remain close, and they remain totally committed to Apple and Moses – as they find their way forward as individuals, they will naturally be more public about seeing other people, if not showy about it.”
The actress and Coldplay frontman, who split last spring, continue to co-parent their children (daughter Apple, 10, and son Moses, 9) and even still vacation together as a family, as recently as last week on a spring break getaway in Mexico.
While family is their main focus, both Paltrow and Martin have found time for romance as well. The actress, 42, has been linked with Glee co-creator Brad Falchuk for over a year, and the pair made their relationship public over the weekend, getting cozy at Robert Downey Jr.’s 50th birthday party.
Meanwhile, the rocker, 38, has been seeing Jennifer Lawrence since September. The under-the-radar duo has been maintaining the relationship from opposite coasts, as the Oscar winner, 25, has been in Boston shooting an upcoming movie while Martin remains in Los Angeles.
Despite the long distance, a source has said they “communicate daily” and pencil in some time for romance, spending Easter weekend together in New York City and then Boston.
“They are both very private and very aware that who they date attracts attention,” the source adds of Martin and Paltrow. “They want to make sure, as their kids are still in elementary school and become aware of their celebrity, that they are not exposed to things that might be hurtful.”
And the source insists there’s no bad blood over taking their new romances public:
“Between the two of them, there is no issue,” the source says. “They are good friends who respect one another deeply as parents. They support one another in that unconditionally.”
Paltrow and Martin may be moving on with other people, but their children are always considered before making any moves.
“One thing that is clear is that the happiness of their two kids is hugely important to them,” the source says. “They make a point of being present as parents, being involved in playdates and parties, making sure their kids are happy where they are.”
For more on Chris Martin, Jennifer Lawrence, Gwyneth Paltrow and Brad Falchuk, pick up the new issue of PEOPLE, on newsstands Friday